Find the Derivative - d/dS0 S_0((3x^3-x^2+2x-4)dx)/( square root of x^2-3x+2)

$S_{0} \frac{(3 x^{3} - x^{2} + 2 x - 4) d x}{\sqrt{x^{2} - 3 x + 2}}$

Factor $x^{2} - 3 x + 2$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $- 3$ .

$- 2, - 1$

Write the factored form using these integers.

$\frac{d}{d S0} [S_{0} \frac{(3 x^{3} - x^{2} + 2 x - 4) d x}{\sqrt{(x - 2) (x - 1)}}]$

Multiply $\frac{(3 x^{3} - x^{2} + 2 x - 4) d x}{\sqrt{(x - 2) (x - 1)}}$ by $\frac{\sqrt{(x - 2) (x - 1)}}{\sqrt{(x - 2) (x - 1)}}$ .

$\frac{d}{d S0} [S_{0} (\frac{(3 x^{3} - x^{2} + 2 x - 4) d x}{\sqrt{(x - 2) (x - 1)}} \cdot \frac{\sqrt{(x - 2) (x - 1)}}{\sqrt{(x - 2) (x - 1)}})]$

Combine fractions.

Combine and simplify the denominator.

Multiply $\frac{(3 x^{3} - x^{2} + 2 x - 4) d x}{\sqrt{(x - 2) (x - 1)}}$ by $\frac{\sqrt{(x - 2) (x - 1)}}{\sqrt{(x - 2) (x - 1)}}$ .

$\frac{d}{d S0} [S_{0} \frac{(3 x^{3} - x^{2} + 2 x - 4) d x \sqrt{(x - 2) (x - 1)}}{\sqrt{(x - 2) (x - 1)} \sqrt{(x - 2) (x - 1)}}]$

Raise $\sqrt{(x - 2) (x - 1)}$ to the power of $1$ .

$\frac{d}{d S0} [S_{0} \frac{(3 x^{3} - x^{2} + 2 x - 4) d x \sqrt{(x - 2) (x - 1)}}{{\sqrt{(x - 2) (x - 1)}}^{1} \sqrt{(x - 2) (x - 1)}}]$

Raise $\sqrt{(x - 2) (x - 1)}$ to the power of $1$ .

$\frac{d}{d S0} [S_{0} \frac{(3 x^{3} - x^{2} + 2 x - 4) d x \sqrt{(x - 2) (x - 1)}}{{\sqrt{(x - 2) (x - 1)}}^{1} {\sqrt{(x - 2) (x - 1)}}^{1}}]$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d}{d S0} [S_{0} \frac{(3 x^{3} - x^{2} + 2 x - 4) d x \sqrt{(x - 2) (x - 1)}}{{\sqrt{(x - 2) (x - 1)}}^{1 + 1}}]$

Add $1$ and $1$ .

$\frac{d}{d S0} [S_{0} \frac{(3 x^{3} - x^{2} + 2 x - 4) d x \sqrt{(x - 2) (x - 1)}}{{\sqrt{(x - 2) (x - 1)}}^{2}}]$

Rewrite ${\sqrt{(x - 2) (x - 1)}}^{2}$ as $(x - 2) (x - 1)$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(x - 2) (x - 1)}$ as ${((x - 2) (x - 1))}^{\frac{1}{2}}$ .

$\frac{d}{d S0} [S_{0} \frac{(3 x^{3} - x^{2} + 2 x - 4) d x \sqrt{(x - 2) (x - 1)}}{{({((x - 2) (x - 1))}^{\frac{1}{2}})}^{2}}]$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{d}{d S0} [S_{0} \frac{(3 x^{3} - x^{2} + 2 x - 4) d x \sqrt{(x - 2) (x - 1)}}{{((x - 2) (x - 1))}^{\frac{1}{2} \cdot 2}}]$

Combine $\frac{1}{2}$ and $2$ .

$\frac{d}{d S0} [S_{0} \frac{(3 x^{3} - x^{2} + 2 x - 4) d x \sqrt{(x - 2) (x - 1)}}{{((x - 2) (x - 1))}^{\frac{2}{2}}}]$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{d}{d S0} [S_{0} \frac{(3 x^{3} - x^{2} + 2 x - 4) d x \sqrt{(x - 2) (x - 1)}}{{((x - 2) (x - 1))}^{\frac{2}{2}}}]$

Rewrite the expression.

$\frac{d}{d S0} [S_{0} \frac{(3 x^{3} - x^{2} + 2 x - 4) d x \sqrt{(x - 2) (x - 1)}}{{((x - 2) (x - 1))}^{1}}]$

Simplify.

$\frac{d}{d S0} [S_{0} \frac{(3 x^{3} - x^{2} + 2 x - 4) d x \sqrt{(x - 2) (x - 1)}}{(x - 2) (x - 1)}]$

Reorder factors in $\frac{(3 x^{3} - x^{2} + 2 x - 4) d x \sqrt{(x - 2) (x - 1)}}{(x - 2) (x - 1)}$ .

$\frac{d}{d S0} [S_{0} \frac{d x \sqrt{(x - 2) (x - 1)} (3 x^{3} - x^{2} + 2 x - 4)}{(x - 2) (x - 1)}]$

Combine $S_{0}$ and $\frac{d x \sqrt{(x - 2) (x - 1)} (3 x^{3} - x^{2} + 2 x - 4)}{(x - 2) (x - 1)}$ .

$\frac{d}{d S0} [\frac{S_{0} (d x \sqrt{(x - 2) (x - 1)} (3 x^{3} - x^{2} + 2 x - 4))}{(x - 2) (x - 1)}]$

Since $\frac{S_{0} d x \sqrt{(x - 2) (x - 1)} (3 x^{3} - x^{2} + 2 x - 4)}{(x - 2) (x - 1)}$ is constant with respect to $S0$ , the derivative of $\frac{S_{0} d x \sqrt{(x - 2) (x - 1)} (3 x^{3} - x^{2} + 2 x - 4)}{(x - 2) (x - 1)}$ with respect to $S0$ is $0$ .

$0$

Find the Derivative - d/dS0 S_0((3x^3-x^2+2x-4)dx)/( square root of x^2-3x+2)

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